Unique prime
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In mathematics, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique iff there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q. Unique primes were first described by Samuel Yates in 1980.
It can be shown that a prime p is of unique period n iff there exists a natural number c such that
- <math>\frac{\Phi_n(10)}{\gcd(\Phi_n(10),n)} = p^c<math>
where Φn(x) is the n-th cyclotomic polynomial; until today, 18 unique primes are known, and no others exist below 1050. The following table gives an overview of all known unique primes Template:OEIS and their periods Template:OEIS:
| Period length | Prime |
|---|---|
| 1 | 3 |
| 2 | 11 |
| 3 | 37 |
| 4 | 101 |
| 10 | 9,091 |
| 12 | 9,901 |
| 9 | 333,667 |
| 14 | 909,091 |
| 24 | 99,990,001 |
| 36 | 999,999,000,001 |
| 48 | 9,999,999,900,000,001 |
| 38 | 909,090,909,090,909,091 |
| 19 | 1,111,111,111,111,111,111 |
| 23 | 11,111,111,111,111,111,111,111 |
| 39 | 900,900,900,900,990,990,990,991 |
| 62 | 909,090,909,090,909,090,909,090,909,091 |
| 120 | 100,009,999,999,899,989,999,000,000,010,001 |
| 150 | 10,000,099,999,999,989,999,899,999,000,000,000,100,001 |
External links
- The Prime Glossary: Unique prime (http://primes.utm.edu/glossary/page.php?sort=UniquePrime)fr:Nombre premier unique